Optimal. Leaf size=59 \[ \frac{3 a^2 \left (a+b \sqrt [3]{x}\right )^{16}}{16 b^3}+\frac{\left (a+b \sqrt [3]{x}\right )^{18}}{6 b^3}-\frac{6 a \left (a+b \sqrt [3]{x}\right )^{17}}{17 b^3} \]
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Rubi [A] time = 0.0473349, antiderivative size = 59, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {190, 43} \[ \frac{3 a^2 \left (a+b \sqrt [3]{x}\right )^{16}}{16 b^3}+\frac{\left (a+b \sqrt [3]{x}\right )^{18}}{6 b^3}-\frac{6 a \left (a+b \sqrt [3]{x}\right )^{17}}{17 b^3} \]
Antiderivative was successfully verified.
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Rule 190
Rule 43
Rubi steps
\begin{align*} \int \left (a+b \sqrt [3]{x}\right )^{15} \, dx &=3 \operatorname{Subst}\left (\int x^2 (a+b x)^{15} \, dx,x,\sqrt [3]{x}\right )\\ &=3 \operatorname{Subst}\left (\int \left (\frac{a^2 (a+b x)^{15}}{b^2}-\frac{2 a (a+b x)^{16}}{b^2}+\frac{(a+b x)^{17}}{b^2}\right ) \, dx,x,\sqrt [3]{x}\right )\\ &=\frac{3 a^2 \left (a+b \sqrt [3]{x}\right )^{16}}{16 b^3}-\frac{6 a \left (a+b \sqrt [3]{x}\right )^{17}}{17 b^3}+\frac{\left (a+b \sqrt [3]{x}\right )^{18}}{6 b^3}\\ \end{align*}
Mathematica [A] time = 0.0454744, size = 41, normalized size = 0.69 \[ \frac{\left (a+b \sqrt [3]{x}\right )^{16} \left (a^2-16 a b \sqrt [3]{x}+136 b^2 x^{2/3}\right )}{816 b^3} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.006, size = 165, normalized size = 2.8 \begin{align*} x{a}^{15}+{\frac{{b}^{15}{x}^{6}}{6}}+91\,{a}^{3}{b}^{12}{x}^{5}+{\frac{45\,b{a}^{14}}{4}{x}^{{\frac{4}{3}}}}+63\,{b}^{2}{a}^{13}{x}^{5/3}+{\frac{455\,{b}^{3}{x}^{2}{a}^{12}}{2}}+585\,{b}^{4}{a}^{11}{x}^{7/3}+{\frac{9009\,{b}^{5}{a}^{10}}{8}{x}^{{\frac{8}{3}}}}+{\frac{5005\,{a}^{9}{b}^{6}{x}^{3}}{3}}+{\frac{3861\,{b}^{7}{a}^{8}}{2}{x}^{{\frac{10}{3}}}}+1755\,{b}^{8}{a}^{7}{x}^{11/3}+{\frac{5005\,{b}^{9}{x}^{4}{a}^{6}}{4}}+693\,{b}^{10}{a}^{5}{x}^{13/3}+{\frac{585\,{b}^{11}{a}^{4}}{2}{x}^{{\frac{14}{3}}}}+{\frac{315\,{b}^{13}{a}^{2}}{16}{x}^{{\frac{16}{3}}}}+{\frac{45\,{b}^{14}a}{17}{x}^{{\frac{17}{3}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.967139, size = 63, normalized size = 1.07 \begin{align*} \frac{{\left (b x^{\frac{1}{3}} + a\right )}^{18}}{6 \, b^{3}} - \frac{6 \,{\left (b x^{\frac{1}{3}} + a\right )}^{17} a}{17 \, b^{3}} + \frac{3 \,{\left (b x^{\frac{1}{3}} + a\right )}^{16} a^{2}}{16 \, b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.44062, size = 432, normalized size = 7.32 \begin{align*} \frac{1}{6} \, b^{15} x^{6} + 91 \, a^{3} b^{12} x^{5} + \frac{5005}{4} \, a^{6} b^{9} x^{4} + \frac{5005}{3} \, a^{9} b^{6} x^{3} + \frac{455}{2} \, a^{12} b^{3} x^{2} + a^{15} x + \frac{9}{136} \,{\left (40 \, a b^{14} x^{5} + 4420 \, a^{4} b^{11} x^{4} + 26520 \, a^{7} b^{8} x^{3} + 17017 \, a^{10} b^{5} x^{2} + 952 \, a^{13} b^{2} x\right )} x^{\frac{2}{3}} + \frac{9}{16} \,{\left (35 \, a^{2} b^{13} x^{5} + 1232 \, a^{5} b^{10} x^{4} + 3432 \, a^{8} b^{7} x^{3} + 1040 \, a^{11} b^{4} x^{2} + 20 \, a^{14} b x\right )} x^{\frac{1}{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 4.30683, size = 207, normalized size = 3.51 \begin{align*} a^{15} x + \frac{45 a^{14} b x^{\frac{4}{3}}}{4} + 63 a^{13} b^{2} x^{\frac{5}{3}} + \frac{455 a^{12} b^{3} x^{2}}{2} + 585 a^{11} b^{4} x^{\frac{7}{3}} + \frac{9009 a^{10} b^{5} x^{\frac{8}{3}}}{8} + \frac{5005 a^{9} b^{6} x^{3}}{3} + \frac{3861 a^{8} b^{7} x^{\frac{10}{3}}}{2} + 1755 a^{7} b^{8} x^{\frac{11}{3}} + \frac{5005 a^{6} b^{9} x^{4}}{4} + 693 a^{5} b^{10} x^{\frac{13}{3}} + \frac{585 a^{4} b^{11} x^{\frac{14}{3}}}{2} + 91 a^{3} b^{12} x^{5} + \frac{315 a^{2} b^{13} x^{\frac{16}{3}}}{16} + \frac{45 a b^{14} x^{\frac{17}{3}}}{17} + \frac{b^{15} x^{6}}{6} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.17465, size = 221, normalized size = 3.75 \begin{align*} \frac{1}{6} \, b^{15} x^{6} + \frac{45}{17} \, a b^{14} x^{\frac{17}{3}} + \frac{315}{16} \, a^{2} b^{13} x^{\frac{16}{3}} + 91 \, a^{3} b^{12} x^{5} + \frac{585}{2} \, a^{4} b^{11} x^{\frac{14}{3}} + 693 \, a^{5} b^{10} x^{\frac{13}{3}} + \frac{5005}{4} \, a^{6} b^{9} x^{4} + 1755 \, a^{7} b^{8} x^{\frac{11}{3}} + \frac{3861}{2} \, a^{8} b^{7} x^{\frac{10}{3}} + \frac{5005}{3} \, a^{9} b^{6} x^{3} + \frac{9009}{8} \, a^{10} b^{5} x^{\frac{8}{3}} + 585 \, a^{11} b^{4} x^{\frac{7}{3}} + \frac{455}{2} \, a^{12} b^{3} x^{2} + 63 \, a^{13} b^{2} x^{\frac{5}{3}} + \frac{45}{4} \, a^{14} b x^{\frac{4}{3}} + a^{15} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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